Electric circuits 2 (XE31EO2)

More difficult problems for the second part of the exam from this subject

 

Problems covers lectured subjects in following themes:

  1. Periodic steady state.
  2. Transient analysis (in time domain). For transient analysis problems the selection of the method is arbitrary, so problems are identical for both time domain and operational analysis.
  3. (Operational) analysis of the transients.
  4. Transfer characteristics of the linear circuits.
  5. Distributed circuits - lossless transmission line.

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  1. Periodic steady state

    1. Circuits shown in figures in the following table are supplied by the periodic non-sinusoidal voltage

      and they are in the steady state. Find the output voltage u 2(t) and its effective (rms) value U 2.

      problem 1



      solution       		problem 2



      solution
      problem 3



      solution       		problem 4



      solution
      problem 5



      solution       		problem 6



      solution
      problem 7



      solution       		problem 8



      solution
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    2. Circuits shown in figures in the following table are supplied by the periodic non-sinusoidal current

      and they are in the steady state. Find the output voltage u2(t) and its effective (rms) value U2.

      problem 1



      solution       		problem 2



      solution
      problem 3



      solution       		problem 4



      solution
      problem 5



      solution       		problem 6



      solution
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    3. Circuits shown in figures in the following table are supplied by the periodic non-sinusoidal voltage u 1(t) with the waveform shown on the figure a) or b) and are in the steady state. Find the output voltage u2(t) and its effective (rms) value U 2.
      Use first three non zero harmonics components of the Fourier series (including a DC component, if that is not zero).
      obrázek a) obrázek b) Pomůcka:
      Standardní obdélníkový průběh posunutý v amplitudě Standardní obdélníkový průběh posunutý v amplitudě a umístěný souměrně k ose napětí Standardní obdélníkový průběh s nulovou ss. složkou
      (Standart square wave advanced by amplitude) (Standart square wave advanced by amplitude and placed symetrically to the voltage axis) CENTER

      problem 1



      solution       		problem 2



      solution
      problem 3



      solution       		problem 4



      solution
      problem 5



      solution       		problem 6



      solution
      problem 7



      solution       		problem 8



      solution
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    4. Circuits shown in figures in the following table are supplied by the periodic non-sinusoidal voltage source .
      The circuits contains ideal voltage controlled voltage source uv = K ur and they are in the steady state. Compute the output current waveform i 2(t) and its effective value I 2. Compute the active power, reactive power, apparent power and deformation power and the power factor of the output part of the circuit.

      problem 1



      solution       		problem 2



      solution
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    5. Circuits shown in figures in the following table are supplied by the periodic non-sinusoidal voltage source
      .
      The circuits contains ideal current controlled voltage source  uv = R ir and they are in the steady state. Compute the output current waveform i2(t) and its effective value I2. Compute the active power, reactive power, apparent power and deformation power and the power factor of the output part of the circuit.

      problem 1



      solution       		problem 2



      solution
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    6. Circuits shown in figures in the following table are supplied by the periodic non-sinusoidal voltage source
      .
      The circuits contains ideal current controlled current source  iv = H ir and they are in the steady state. Compute the output voltage waveform u 2(t) and its effective value U 2. Compute the active power, reactive power, apparent power and deformation power and the power factor of the output part of the circuit.

      problem 1



      solution       		problem 2



      solution
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    7. Circuits shown in figures in the following table are supplied by the periodic non-sinusoidal voltage source
      and they are in the steady state. Compute waveform of the output voltage u 2(t) and its effective value U 2. Compute the active power, reactive power, apparent power and deformation power and the power factor of the voltage source.

      problem 1



      solution       		problem 2



      solution
      problem 3



      solution       		problem 4



      solution
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    8. Circuits shown in figures in the following table are supplied by the periodic non-sinusoidal current source
      and they are in the steady state. Compute waveform of the output voltage u2(t) and its effective value U2. Compute the active power, reactive power, apparent power and deformation power and the power factor of the current source.

      problem 1



      solution       		problem 2



      solution
      problem 3



      solution       		problem 4



      solution
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    9. Given the circuit in Fig. 1 in the table below that is excited by the periodic non-sinusoidal voltage u(t ) with waveform according to Fig. 2. The circuit is in the steady state and at resonance for the first harmonic component of the exciting voltage. Evaluate the necessary quality factor 

      to obtain the
      1. current i(t ),
      2. voltage uc(t)
      (in the question in the exam will be always only one of the options) where amplitude of any higher harmonic component will be less than x % of the amplitude of the first harmonic component.

      Fig. 1







      solution       		Fig. 2
      Standardní obdélníkový průběh s nulovou ss.složkou
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    10. Given the circuit according the Fig. 1 in the table below that is excited by the periodic non-sinusoidal current i(t ) with waveform according to Fig. 2. The circuit is in the steady state and at resonance for the first harmonic component of the the exciting current. Evaluate the necessary quality factor 

      to obtain the
      1. voltage u(t),
      2. current ic(t)
      (in the question in the exam will be always only one of the options) where amplitude of any higher harmonic component will be less than x % of the amplitude of the first harmonic component.

      Fig. 1







      solution       		Fig. 2
      Standardní obdélníkový průběh s nulovou ss.složkou
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    11. Given circuit according to the Fig. 1 that is excited by the periodic non-sinusoidal voltage source u(t ) of the waveform
      u(t) = U0 + Um1 sin( ω0t)  + Um2 sin(2 ω0t)  + Um3 sin(3 ω0t) [V].
      Evaluate the waveform of the current i(t ) in the steady state and the active power of the source.

      Fig. 1



      solution
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    12. Given circuit according to the Fig. 1 that is excited by the periodic non-sinusoidal current i(t ) of the waveform
      i(t) = I0 + Im1 sin( ω0t)  + Im2 sin(2 ω0t)  + Im3 sin(3 ω0t) [A].
      Evaluate the waveform of the voltage u(t ) in the steady state and active power of the source.

      Fig. 1



      solution
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  2. Transient analysis (in time domain).

    1. Given circuits according to the figures in table below have been in the steady state for time t < 0. Find the waveform of the circuit variable shown on the figure (voltage ux(t), or current ix(t)) both for t > 0 and t < 0. Draw founded waveform in scale on both axes. The waveform of the shown circuit variable find both for case, when the switch is switched on at t = 0, and for case, when the swith is switched off.

      problem 1



      solution       		problem 2



      solution
      problem 3



      solution       		problem 4



      solution
      problem 5



      solution       		problem 6



      solution
      problem 7




      solution       		problem 8



      solution
      problem 9



      solution       		problem 10



      solution
      problem 11



      solution       		problem 12



      solution
      problem 13



      solution       		problem 14



      solution
      problem 15



      solution
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    2. The circuit in figure has been in the steady state for t < 0. Find the waveform of the current ix(t) both for t > 0 and t < 0. Draw found waveform in scale on both axes. The waveform of the current find both in the case, when the swith is switched from the position 1 to the position 2 at t = 0, and for the opposite case.

      problem 1



      solution
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    3. Given circuit according to the figure has been in the steady state for t < 0. Find the waveform of the voltage ux(t) both for t > 0 and t < 0. Draw found waveform in scale on both axes. The waveform of the voltage find both for case, when the switch is switched on at t = 0, and for case, when the swith is switched off.

      problem 1



      solution
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    4. Given circuits according to the figures in the table below have been in the steady state for t < 0. Find the waveform of the circuit variable shown on the figure (voltage ux(t), or current ix(t)) both for t > 0 and t < 0. Draw founded waveform in scale on both axes. The waveform of the shown circuit variable find both for case, when the switch is switched from the position 1 to the position 2 at t = 0, and for opposite case.

      problem 1



      solution       		problem 2



      solution
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    5. Given circuits according to the figures in the table below have been in the steady state for t < 0. Find the waveform of the circuit variable shown on the figure (voltage ux(t), or current ix(t)) both for t > 0 and t < 0. Draw founded waveform in scale on both axes. The waveform of the shown circuit variable find both for case, when the switch is switched on and off at t = 0.

      problem 1



      solution       		problem 2



      solution
      problem 3



      solution       		problem 4



      solution
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    6. The circuit in figure has been in the steady state for t < 0. Find the waveform of the voltage ux(t) both for t > 0 and t < 0, if the switch is switched on at time t = 0. Draw founded waveform in scale on both axes.
      Recommendation: Use Laplace transform for the solution.

      problem 1



      solution
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  3. (Operational) analysis of the transients

    Problems are identical to the previous theme, as used method of the solution is not imposed and depends only on your will.
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  4. Transfer characteristics of linear circuits

    For circuits according to the table below evaluate:

    1. Transfer function P(p) = U2(p) / U1(p) and write it in the form suitable for drawing of the frequency responses. Draw the asymptotic magnitude and phase frequency responses in logarithmics coordinates and mark break frequencies and constant values (of the horizontal asymptots).
    2. Unit step response a(t) and sketch it.
    3. Unit impulse response w(t) and sketch it.
    The values of the circuit elements are given in the table below under the corresponding figure.
    Note: In the written part of the exam will be required calculation of one from this options only, that is either transfer function, or unit step response, or unit impulse response.

    problem 1





    solutions: P(p), a(t), w(t)
    		problem 2





    solutions: P(p), a(t), w(t)
    problem 3





    solutions: P(p), a(t), w(t)
    		problem 4





    solutions: P(p), a(t), w(t)
    problem 5





    solutions: P(p), a(t), w(t)
    		problem 6





    solutions: P(p), a(t), w(t)
    problem 7





    solutions: P(p), a(t), w(t)
    		problem 8





    solutions: P(p), a(t), w(t)
    problem 9





    solutions: P(p), a(t), w(t)
    		problem 10





    solutions: P(p), a(t), w(t)
    problem 11





    solutions: P(p), a(t), w(t)
    		problem 12





    solutions: P(p), a(t), w(t)
    problem 13





    solutions: P(p), a(t), w(t)
    		problem 14





    solutions: P(p), a(t), w(t)
    problem 15





    solutions: P(p), a(t), w(t)
    		problem 16





    solutions: P(p), a(t), w(t)
    problem 17





    solutions: P(p), a(t), w(t)
    		problem 18





    solutions: P(p), a(t), w(t)
    problem 19





    solutions: P(p), a(t), w(t)
    		problem 20





    solutions: P(p), a(t), w(t)
    problem 21





    solutions: P(p), a(t), w(t)
    		problem 22





    solutions: P(p), a(t), w(t)
    problem 23





    solutions: P(p), a(t), w(t)
    		problem 24





    solutions: P(p), a(t), w(t)
    problem 25





    solutions: P(p), a(t), w(t)
    		problem 26





    solutions: P(p), a(t), w(t)
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  5. Transients on lossless transmission line

  1. Draw the elementary section of a homogeneous lossless transmission line, deduce its basic line equation for voltage and write its solution. The homogeneous lossless transmission line of the length l with parameters L, C is short-circuited. The DC voltage source Ui0 with internal resistance Ri = R0 is connected in the time instant t = 0 to the line. Draw the voltage waveforms u1(t), us(t) and u2(t) at the beginning, in the middle and at the end of the line.


    solution


  2. Prove, that wave equations are satisfied by the arbitrary function u(x,t) = f1(x-vt) + f2(x+vt) for lossless transmission line. The homogeneous lossless transmission line of the length l with parameters L, C is open-circuited. The DC voltage source Ui0 with internal resistance Ri = R0 is connected in the time instant t = 0 to the line. Draw the voltage waveforms u1(t), us(t) and u2(t) at the beginning, in the middle and at the end of the line.


    solution


  3. Draw the elementary section of a homogeneous lossless transmission line, deduce its basic line equation for voltage and write its solution. The homogeneous lossless transmission line of the length l with parameters L, C is loaded by the resistor Rs. The DC voltage source Ui0  with internal resistance Ri = R0 is connected in the time instant t = 0 to the line. Draw the voltage waveforms u1(t), us(t) and u2(t) at the beginning, in the middle and at the end of the line.


    solution


  4. Draw the elementary section of a homogeneous lossless transmission line, deduce its basic line equation for current and write its solution. The homogeneous lossless transmission line of the length l with parameters L, C is short-circuited. The DC voltage source Ui0  with internal resistance Ri = R0 is connected in the time instant t = 0 to the line. Draw the current waveforms i1(t), is(t) and i2(t) at the beginning, in the middle and at the end of the line.


    solution


  5. Prove, that wave equations are satisfied by the arbitrary function u(x,t) = f1(x-vt) + f2(x+vt) for lossless transmission line. The homogeneous lossless transmission line of the length l with parameters L, C is open-circuited. The DC voltage source Ui0  with internal resistance Ri = R0 is connected in the time instant t = 0 to the line. Draw the current waveforms i1(t), is(t) and i2(t) at the beginning, in the middle and at the end of the line.


    solution


  6. Prove, that wave equations are satisfied by the arbitrary function u(x,t) = f1(x-vt) + f2(x+vt) for lossless transmission line. The homogeneous lossless transmission line of the length l with parameters L, C is loaded by the resistor Rs. The DC voltage source Ui0 with internal resistance Ri = R0 is connected in the time instant t = 0 to the line. Draw the current waveforms i1(t), is(t) and i2(t) at the beginning, in the middle and at the end of the line.


    solution


  7. Deduce formulas for transmission and reflection coefficient for the voltage waves at the point of connection of two lossless transmission lines with different wave resistances R01, R02. Lossless two parallel transmission line in the air space of the length l1 and wave resistance R01 is connected to the cabel line with the length l2 with the wave resistance R02 and the relative permitivity of the insulating material is er. To the input of the first line is connected the DC voltage source Ui0 with internal resistance Ri = R01. Cabel line is terminated by the resistor Rs =  R02. Draw the voltage waveforms u1(t), us(t) and u2(t) at the beginning of the first line, at the point of connection of both lines and at the end of the cabel line, supposing the cable line is also lossless.


    solution


  8. Deduce formulas for transmission and reflection coefficient for the voltage waves at the point of connection of two lossless transmission lines with different wave resistances R01, R02. The homogeneous lossless transmission line of the length l with parameters L, C is open-circuited. The DC voltage source Ui0 with zero internal resistance Ri is connected in the time instant t = 0 to the line. Draw the voltage waveforms u1(t), us(t) and u2(t) at the beginning, in the middle and at the end of the line at the time tc.


    solution


  9. Deduce formulas for transmission and reflection coefficient for the voltage waves at the point of connection of two lossless transmission lines with different wave resistances R01, R02. The homogeneous lossless transmission line of the length l with parameters L, C is loaded by the resistor Rs. The DC voltage source Ui0 with zero internal resistance Ri is connected in the time instant t = 0 to the line. Draw the voltage waveforms u1(t), us(t) and u2(t) at the beginning, in the middle and at the end of the line at the time tc.


    solution


  10. Draw the elementary section of a homogeneous lossless transmission line, deduce its basic line equation for current and write its solution. The homogeneous lossless transmission line of the length l with parameters L, C is short-circuited. The DC current source Ii0 with internal conductance Gi = 0 S is connected in the time instant t = 0 to the line. Draw the current waveforms i1(t), is(t) and i2(t) at the beginning, in the middle and at the end of the line at the time tc.


    solution


  11. Draw the elementary section of a homogeneous lossless transmission line, deduce its basic line equation for current and write its solution. The homogeneous lossless transmission line of the length l with parameters L, C is loaded by the resistor Rs. The DC current source Ii0 with internal conductance Gi = 0 S is connected in the time instant t = 0 to the line. Draw the current waveforms i1(t), is(t) and i2(t) at the beginning, in the middle and at the end of the line at the time tc.


    solution


  12. Draw the elementary section of a homogeneous lossless transmission line, deduce its basic line equation for voltage and write its solution. The homogeneous lossless transmission line of the length l with parameters L, C is short-circuited. The line is supplied by the voltage source ui(t) with the internal resistor Ri = R0, which send into the line one very short rectangular pulse of the amplitude Ui of the width t0. Draw the voltage waveforms u1(t), us(t) and u2(t) at the beginning, in the middle and at the end of the line.


    solution


  13. Prove, that wave equations are satisfied by the arbitrary function u(x,t) = f1(x-vt) + f2(x+vt) for lossless transmission line. The homogeneous lossless transmission line of the length l with parameters L, C is open-circuited. The line is supplied by the voltage source ui(t) with the internal resistor Ri = R0, which send into the line one very short rectangular pulse of the amplitude Ui and of the width t0. Draw the voltage waveforms u1(t), us(t) and u2(t) at the beginning, in the middle and at the end of the line.


    solution


  14. Draw the elementary section of a homogeneous lossless transmission line, deduce its basic line equation for voltage and write its solution. The homogeneous lossless transmission line of the length l with parameters L, C is loaded by the resistor Rs. The line is supplied by the voltage source ui(t) with the internal resistor Ri = R0, which send into the line one very short rectangular pulse of the amplitude Ui and of the width t0. Draw the current waveforms u1(t), us(t) and u2(t) at the beginning, in the middle and at the end of the line.


    solution


  15. Draw the elementary section of a homogeneous lossless transmission line, deduce its basic line equation for current and write its solution. The homogeneous lossless transmission line of the length l with parameters L, C is short-circuited. The line is supplied by the voltage source ui(t) with the internal resistor Ri = R0, which send into the line one very short rectangular pulse of the amplitude Ui and of the width t0. Draw the current waveforms i1(t), is(t) and i2(t) at the beginning, in the middle and at the end of the line.


    solution


  16. Prove, that wave equations are satisfied by the arbitrary function u(x,t) = f1(x-vt) + f2(x+vt) for lossless transmission line. The homogeneous lossless transmission line of the length l with parameters L, C is open-circuited. The line is supplied by the voltage source ui(t) with the internal resistor Ri = R0, which send into the line one very short rectangular pulse of the amplitude Ui and of the width t0. Draw the current waveforms i1(t), is(t) and i2(t) at the beginning, in the middle and at the end of the line.


    solution


  17. Prove, that wave equations are satisfied by the arbitrary function u(x,t) = f1(x-vt) + f2(x+vt) for lossless transmission line. The homogeneous lossless transmission line of the length l with parameters L, C is loaded by the resistor Rs. The line is supplied by the voltage source ui(t) with the internal resistor Ri = R0, which send into the line one very short rectangular pulse of the amplitude Ui and of the width t0. Draw the current waveforms i1(t), is(t) and i2(t) at the beginning, in the middle and at the end of the line.


    solution


  18. Deduce formulas for transmission and reflection coefficient for the voltage waves at the point of connection of two lossless transmission lines with different wave resistances R01, R02. Lossless two parallel transmission line in the air space of the length l1 and wave resistance R01 is connected to the cabel line with the length l2 with the wave resistance R02 and the relative permitivity of the insulating material is er. The line is supplied by the voltage source ui(t) with the internal resistor Ri = R01, which send into the line one very short rectangular pulse of the amplitude Ui and of the width t0. Cabel line is terminated by the resistor Rs = R02 W. Draw the voltage waveforms u1(t), us(t) and u2(t) at the beginning of the first line, at the point of connection of both lines and at the end of the cabel line, supposing the cable line is also lossless.


    solution


  19. Deduce formulas for transmission and reflection coefficient for the voltage waves at the point of connection of two lossless transmission lines with different wave resistances R01, R02. The homogeneous lossless transmission line of the length l with parameters L, C is open-circuited. The line is supplied by the voltage source ui(t) with the zero internal resistance, which send into the line one very short rectangular pulse of the amplitude Ui and of the width t0. Draw the voltage waveforms u1(t), us(t) and u2(t) at the beginning, in the middle and at the end of the line at the time 0 till tc.


    solution


  20. Deduce formulas for transmission and reflection coefficient for the voltage waves at the point of connection of two lossless transmission lines with different wave resistances R01, R02. The homogeneous lossless transmission line of the length l with parameters L, C is short-circuited. The line is supplied by the voltage source ui(t) with the zero internal resistance, which send into the line one very short rectangular pulse of the amplitude Ui and of the width t0. Draw the voltage waveforms u1(t), us(t) and u2(t) at the beginning, in the middle and at the end of the line at the time 0 till tc.


    solution


  21. Deduce formulas for transmission and reflection coefficient for the voltage waves at the point of connection of two lossless transmission lines with different wave resistances R01, R02. The homogeneous lossless transmission line of the length l with parameters L, C = is loaded by the resistor Rs. The line is supplied by the voltage source ui(t) with the zero internal resistance, which send into the line one very short rectangular pulse of the amplitude Ui and of the width t0. Draw the voltage waveforms u1(t), us(t) and u2(t) at the beginning, in the middle and at the end of the line at the time 0 till tc.


    solution


  22. Draw the elementary section of a homogeneous lossless transmission line, deduce its basic line equation for current and write its solution. The homogeneous lossless transmission line of the length l with parameters L, C is short-circuited. The line is supplied by the ideal current source ii(t) with the zero internal resistance, which send into the line one very short rectangular pulse of the amplitude Ii and of the width t0. Draw the current waveforms i1(t), is(t) and i2(t) at the beginning, in the middle and at the end of the line at the time 0 till tc.


    solution


  23. Draw the elementary section of a homogeneous lossless transmission line, deduce its basic line equation for current and write its solution. The homogeneous lossless transmission line of the length l with parameters L, C is open-circuited. The line is supplied by the ideal current source ii(t) with the zero internal resistance, which send into the line one very short rectangular pulse of the amplitude Ii and of the width t0. Draw the current waveforms i1(t), is(t) and i2(t) at the beginning, in the middle and at the end of the line at the time 0 till tc.


    solution


  24. Draw the elementary section of a homogeneous lossless transmission line, deduce its basic line equation for current and write its solution. The homogeneous lossless transmission line of the length l with parameters L, C is loaded by the resistor Rs. The line is supplied by the ideal current source ii(t) with the zero internal resistance, which send into the line one very short rectangular pulse of the amplitude Ii and of the width t0. Draw the current waveforms i1(t), is(t) and i2(t) at the beginning, in the middle and at the end of the line at the time 0 till tc.


    solution


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